Uploaded by YaleCourses on 22.09.2008

Transcript:

Professor Ramamurti Shankar: So,

let's begin now. First of all,

I'm assuming all of you have some idea what special

relativity means. There are two theories of

relativity, one is the special theory and one is the general

theory. The general theory is something

that we won't do in any detail. Special theory is something we

will do in reasonable detail. So, it's good to begin by

asking some of you what is your present understanding of what

the subject is all about. Yes, sir?

The Yale cap, what do you think it's about?

Student: It's about relative speed in two reference

systems. Professor Ramamurti

Shankar: Okay, it's about relative speed in

two reference systems. I'll come to you;

then I'll come to you. Student: It's based on

the postulate that the laws of physics are the same in any two

references moving in uniform motion relative to one another

and the speed of light is constant in all references.

Professor Ramamurti Shankar: Okay,

I will take the last row there. Student: [inaudible]

Professor Ramamurti Shankar: Okay,

so what I've heard so far is that the laws of physics are the

same for two people who are both in inertial frames of reference

and the velocity of light's a constant.

Right. That's certainly the way we

understand the special relativity theory.

But it's a very old one. It's been going on long before

Einstein came. There was a theory of

relativity at the time of Newton and that's where I want to

begin. Relativity is not a new idea at

all, it's an old one. And the old idea can be

illustrated in this way and it will agree with your own

experience. So, the standard technique for

all of relativity is to get these high speed trains.

I'm going to have our own high speed train;

this is the top view of the train.

And like in everything I do, we'll get away with the lowest

number of dimensions, which happens to be this one

spatial dimension and of course there is time.

So, the train is moving along the x axis.

You are in this train. You board the train and all the

blinds are closed because you don't want to look outside.

That's not because you're traveling through some parts of

New Jersey; you don't want to look outside

for this particular experiment. You get into the train,

you settle down and you explore the world around you.

You pour yourself a drink, you play pool,

you juggle some ping pong balls, tennis balls,

and you have a certain awareness of what's happening,

namely, your understanding of the mechanical world,

and then you go to sleep. When you are sleeping,

some unseen hand gives to this train a large velocity,

200 miles an hour. The question is,

"When you wake up, can you tell if you're moving

or not?" That's the whole question.

Will this speed, whatever I gave you,

200 miles per hour--will it do anything to you in this train

that will betray that velocity? So, when you wake up will you

say I'm moving or not? Now, you might say,

I'm not moving because I'm on Amtrak and I know this train is

not going anywhere. That kind of sociological

reason, by the way, there are many of them,

you cannot invoke. You can only say,

"I'm on this train. Is anything different?"

And the claim is that nothing will be different.

You just will not know you are moving.

Now, if the train picks up speed, or slows down,

you will know right away. If it picks up speed or

accelerates, you find yourself pushed against the back of the

seat or if the driver slams on the brake,

you will slam into the front of the seat in front of you.

No one is saying that when the motion is accelerated,

you will not know. Accelerated motion can be

detected in a closed train without looking outside.

The question is uniform velocity, no matter how high,

can that be perceived? Can that be detected?

So, at the time of Galileo and Newton everybody agreed that you

cannot detect it. Remember that if you started

out and Newton's laws worked for you, you are called an inertial

observer. One of the laws you want is,

if you leave something, it should stay where it is.

When the train is accelerating, that won't be true.

You leave things on the floor when it's accelerating,

things will slide backwards. So, with no apparent force

acting on it, things will begin to

accelerate; that's a non-inertial frame.

We are not interested in that. You started out as an observer

for whom the laws of Newton work, the laws of inertia work,

F = ma, then you go to sleep and you wake up.

So, when I said everything looks the same,

I really meant that the laws of Newton continue to be the same

because if the laws of Newton are the same,

everything will look the same. That's what it means to say

"everything looks the same." Our expectations of what

happens when I throw it up or what happens when two billiard

balls collide, everything is connected to the

laws of Newton. So, the claim is,

the laws of Newton will be unchanged when this velocity is

added on to you. Now, we should be clear about

one thing. If there is a train next to you

in the beginning -- let me just put it on this side for

convenience -- and you got in and you boarded this train but

you looked at this train and it was not moving.

If you lift the blind and look through, you'll see the other

train and there's another passenger in the other train and

you look at each other, you're not moving.

When you wake up after this brief nap, you find when you

look outside the other train is moving at 200 miles an hour.

The question is, "Can you tell if it's you who's

responsible for this relative motion,

or maybe nothing happened to you and the other train is

moving the opposite way?" And the claim of relativity is

that you really cannot tell. You can tell there is motion

between the two trains that wasn't there before.

That's very clear if you look outside but there is no way to

tell what actually happened when you were sleeping.

Whether you were given the velocity of 200 to the right or

the other train was given a velocity of 200 to the left or

maybe a combination of the two, you just cannot tell.

That's the word "relative." So far -- I didn't tell you --

if you have only one train, what I told you earlier,

is that uniform velocity does not leave its imprint on

anything you can measure. If you look outside,

of course you can see the motion of the other train,

but you still cannot tell who is moving.

You cannot distinguish between different possibilities.

So, you have every right to insist that you are not moving

and the other train is moving the opposite way.

Once again, you can make this argument only for uniform

relative motion. If your train is accelerating,

now I'm saying it as if it is an absolute thing,

and it is. You cannot say,

"I'm not accelerating, the other train is accelerating

in the other direction." You cannot say that because

you're the one who is barfing up and throwing up and slamming

your head on the wall; nothing is happening to the

other person. You cannot say "I'm still in

the same frame, you are going the opposite

way." If you are going the opposite

way, why am I throwing up? Or if you are in a rocket and

the rocket's taking off and the G forces are enormous,

many times your weight, it is the astronauts who are

going through the discomfort. At that time they cannot say we

are at rest and everyone is going the opposite way because

no one else is in danger, but they are.

So, accelerated motion will produce effects.

You cannot talk your way out of that.

But uniform velocity will produce no effects on you and no

effects on the other person. You can detect relative motion

but you cannot in any sense maintain that you are moving and

he's not or that he's moving and you are not.

You can say, "I am at rest,

things are the same as before, the train is moving the

opposite way." Now, if you go in the Amtrak

and you look outside and you don't see another train,

but you see the landscape, you see trees and cows and

everything, going at 200 miles an hour in the opposite

direction, you have some reason to believe

that probably the ground is not moving and you are moving.

But that's just based on what I called earlier some sociological

factors. In other words,

it's completely possible to devise an experiment in which

somebody puts the whole landscape on wheels and when you

go to sleep the landscapes, cows and trees are made to move

the opposite way. Not very likely,

but that's because we know in practice no one is going to

bother to do that just to fool you.

But if that did happen, you won't know the difference.

So, the reason we rule that out is we know some extraneous

things not connected to the laws of physics.

That's why we don't like to open the window and look at the

landscape because then we have a bias.

Open the window and look at another train and you just won't

know. That is the principle of

relativity, that uniform motion between two observers,

both of whom are inertial, is relative.

Each one can insist that he or she is not moving;

the other person is the one who is moving.

Of course, now, if the two, in reality,

if the two trains were at rest--Let's imagine my train got

accelerated. So, during the time it was

accelerated, I would know, but if I was sleeping at the

time, I don't know and when I wake up

and the acceleration is gone and the velocity is constant,

that's when I say, "I just cannot tell."

All right. Now, let's show once and for

all that the laws of Newton are not going to be modified.

So, you find the laws of Newton before you go to sleep,

you wake up, you find them again,

you'll get the same laws; that's the claim.

I hope you understand that all the mechanical things you see in

the world around you come just from F = ma.

We have seen projectiles and collision of billiard balls and

rockets; they're all Newtonian mechanics.

So, to say that things will look the same is to say the laws

of Newton that you will deduce before and after waking up will

be the same. So, let's show that.

When you show that, you're really done with it once

and for all. So, let's do the following.

Here is the x axis and here is my frame of reference.

This is my x axis. Let's call this the origin.

The frame goes to negative and positive x values.

Pick some object sitting at the point x.

Now, we are going to first define the notion of an event.

An event is something that happens at a certain place at a

certain time; that's called an event.

For example, if there's a little firecracker

going off somewhere at some time, the x is where it

happened and the t is when it happened.

So, this is space-time. Once again, space-time does not

require Einstein coming in at all.

We have known for thousands of years that if you want to set up

a meeting with somebody, you've got to say where and

you've got to say when and things do happen in space-time.

The fact that you need x and t,

or, if you're living in three spatial dimensions,

the fact that you need x, y,

z and t is not new.

That is not the revolution Einstein created.

The fact that you need four coordinates to label an event is

nothing new. What he did that is new will be

clear later. So, does everyone understand

what an event means? Okay?

An event is something that happens and to say exactly where

and when it happened, in our world of one dimension,

we give it an x and we give it a t.

Now, that's me, and I'm going to give my frame

of reference the name S. It turns out S is not

just based on just my name. This is the canonical name for

two observers, one is called S and one

is called S prime. So, S prime,

let's say, is you. So, your frame of reference is

going to be taken to be sliding relative to mine.

So, let's draw a y axis here.

We don't really deal with the y coordinate but just to

give you a feeling, this is my y axis;

that's my x axis; this is my origin.

y is not going to play a big role.

Now, you are sliding this way, to the right,

and your speed or velocity is always denoted by u.

Some number of meters per second, you are zooming to the

right at some speed u. So, imagine now you are going

past me. At some instant--I'm sitting

here at the origin of coordinates, you cross me and

then a little later you are somewhere there.

So, that's your y frame axis;

that's your origin. And the same event you say has

a coordinate x prime.

We arrange it so that when you zoom past me,

you set your clock to zero and I set my clock to zero.

When you want to set the clock to zero is completely arbitrary.

So, we will decide right when you pass me.

I'm at the origin of my coordinates, you're at the

origin of your coordinates. When you pass me,

we'll click our stop watches and we will set the time to

zero. So, here's an event.

You and I crossed. What are the coordinates for

that event? According to me,

that event occurred at x = 0 and the time was chosen to

be zero. According to you,

your origin was also on top of my origin, so x prime was

zero and the time is just the time.

Everybody has a single time, and that time is called zero.

That is one event. We made the coordinate of the

event zero zero for both you and me.

It is zero in space because my origin crossed your origin.

That crossing took place at my origin;

that's why my x is zero and took place at your origin,

that's where your x prime is zero and the common

time we chose to be zero by convention.

Then, we want a second event. So, let's say the second event

is some firecracker going off here.

Here is something I should explain that I used to forget in

the previous years. When I say I am moving,

I imagine I am part of a huge team of people who are all

moving with me. So, I've got agents all over

the x axis who are my eyes and ears;

they are looking out for me. So, even though I am here,

if there's a firecracker exploding here,

my guys will tell me. And you are carrying your own

agents. Let's say at every point

x you have a reporter, x = 1,2,

3,4; there are people sitting and

watching. So, when I say I see something,

I really mean me and my buddies, all traveling the same

train at the same speed, all over space taking notes on

what's happening. We'll pool our information

later but we know this explosion took place here.

I'll simplify it by saying I know an explosion took place

here at location x at time t.

So, this is our crossing. This event is when we crossed.

Then, there is a firecracker. For the firecracker,

I have to give some events [should have said

"coordinates"]. I say it took place at location

x at time t.

What do you say? You measure the distance from

your origin, you call this some x prime,

the time is still t. In Newtonian mechanics,

the time is just the time. How many seconds have passed is

the same for everybody. The question is,

"What is the relation between x prime and x?"

That's what we want to think about.

So, you guys should think before I write down the answer.

What's the relation of x prime to x?

Well, this event took place at time t,

so I know that your origin is to the right by an amount

u times t.

So, the distance from your origin for this event,

I maintain is x prime is x - ut.

Again, I want all of you to follow everything.

These are all simple notions. Our origins coincided at zero

time for the event that occurs at time t.

Therefore, in a sense, you are rushing toward the

event. You've gone a distance

ut. Therefore, the distance from

your origin to the event will be less than mine by this amount

ut. This is the law of

transformation of coordinates in Newtonian mechanics.

If you have an event--if you want, formally,

you can define it time, t prime for the primed

observer. It goes without saying that

t prime and t are the same.

There's no notion of time for me and time for you.

There's universal time in Newtonian mechanics.

It just runs. We can call some time a zero.

Once we have agreed, if you say you and I met at

t = 0 and an explosion took place 5 seconds after our

meeting, it's going to be 5 seconds

after our meeting for me and it's going to be 5 seconds after

the meeting for you. The time difference between two

events is the same for all people.

This is called the Galilean transformation.

What are the consequences of Galilean transformation?

Well, let's look at the fact that x prime is x -

ut.

Remember, everything is varying with time.

So x prime is a function of time and x is a

function of time, if you are watching a moving

particle. Suppose this firecracker is not

just one event, but it's a moving object.

Let's give the object some speed;

it's moving to the right. Then the velocity,

according to me--I'm going to call v as dx/dt is

the velocity. Let's just call it a bullet,

according to S. Then w -- it's the

standard name -- is the velocity of the bullet according to

S prime. So, what I've done is,

I first took one event and I gave it some coordinates and I

told you how to transform the coordinate from one person to

the other person. But now, take that point

x not to be a fixed location but a moving object,

so that as a function of time that body is moving.

Then its velocity at any time is dx prime dt

according to you; that is the dx/dt

according to me, minus the derivative of this,

which is u. Now, does that make sense?

This should agree with common sense.

For example, if that bullet is going at 600

miles per hour to the right, that is 600 for me,

and you are going to the right in your train at 200 miles per

hour, you should measure the bullet speed to be reduced by

200 and you should get 400. That's all it means.

The two people will disagree on the velocity of the bullet

because they are moving relative to each other.

This is the way you will add velocities.

But let's look at the acceleration.

dw/dt is going to be dv/dt - 0 because

u is a constant.

That means you and I agree on the acceleration of the body.

We disagree on where it is. We disagree on how fast the

bullet's moving. But we agree on the

acceleration of the body because all I've done is add a constant

velocity to everything you see. Therefore, if according to you

the velocity of the body is not changing, according to me the

velocity of the body is not changing,

because the constant added will drop out of the difference.

Or, if the body has an acceleration,

we'll both get the same answer for the acceleration.

So, that is the common acceleration a.

So, if you like, a-prime is the same as

a.

So, the acceleration of bodies doesn't change when you go from

one frame of reference to another one going at a constant

speed.

All right, so let's look at F = ma,

which is md^(2)x/dt^(2) is equal to some force on the

body. And you look at the body and

you say d^(2)x prime over dt^(2) is the force on

your body. First, I want to convince you

we wanted to see that the left-hand sides are equal

because the acceleration's the same.

Then, I want to convince you that the right-hand sides are

also going to be equal. I can take many examples but

eventually you will get the point.

Let us not consider one body, but let's consider two bodies.

Two bodies are feeling a certain force due to,

say, gravitation. And gravitation is,

of course, a force in three dimensions but let's write the

force in just one dimension. And let's say the force of

gravity is equal to 1 over x_1 -

x_2. Force on 1 due to 2 and the

force on 2 due to 1 will be minus 1 over x_1 -

x_2. The real forces are separation

in three dimensions but this is a fictitious force.

I want to call it gravity. It is any force that depends on

the coordinates of the two particles.

So, I will say m_1d^(2)x

_1 over dt^(2) is 1 over

x_1 - x_2.

And m_2d^(2)x _2 over

dt^(2) is minus 1 over x_1 minus--I

have forgotten constants like g and

m_1 and m_2.

They don't matter for this purpose.

So, here are two bodies. They feel a force for each

other and I've discovered what the force is.

It's 1 over x_1 - x_2.

I don't care if it's 1 over x_1 -

x_2 or (x_1 -

x_2)^(2); that's not important.

What's important is, it depends on x_1

- x_2. You come along and you study

the same two masses. What will you say is happening?

You will say, m_1d^(2)x

_1 prime over dt^(2) is equal to 1 over

x_1 prime minus x_2 prime.

Maybe I will--I'm sorry. Let me do it a little better.

I can tell you what you will see.

Given this is what I see, I can tell you what you will

see. Let's do that in our head.

We know that the acceleration is the same for any mass so I'm

going to write this thing as m of dx over prime

dt^(2). In other words,

the acceleration according to me is the same as the

acceleration according to you. Then, I'm also going to write

the right-hand side as x_1 prime minus

x_2 prime. Do you understand that?

If there are two bodies feeling a force, if you see it from a

moving train, the distance between the two

bodies is the same for you and me,

because x_1 prime is x_1 -

ut and x_2 prime is x_2 -

ut. Take the difference;

the difference between the location of the particles is the

same for you and me. Acceleration is the same,

mass is postulated to be the same, so I know that you will

get the same law that I get. You will get F = ma;

your acceleration will be the same as mine;

the force you attribute between the two bodies will also be the

same. That is why I know that you

will also deduce the same Newtonian laws that I will.

You can also say it differently. If I woke up from my nap and I

am now in a moving train and I examine the world around me,

I'm going to get the same F = ma.

Because as seen by a person on the ground, the masses obey F

= ma. I am in this moving train now

but I have the same acceleration for each mass and I have the

same force. So, if you want,

I'll complete the second equation.

m_2d^(2) x_2 prime over

dt^(2) will be minus 1 over x_1 prime

minus x_2 prime.

If this is a little difficult, we should talk about this.

I'm telling you that if I deduce F = ma and the

F depends on the separation between the

particles, then I'm sure that you will

find the same laws of motion because the acceleration is the

same that I get because we have seen a-prime is the same

as a. And the force will also be the

same because the force depends on the separation between

particles. And that doesn't depend on

which train you're in or it's not affected by adding a

constant velocity to the frame of reference.

So, if you like, this is the way you prove in

Newtonian mechanics the principle of relativity.

So, not only is it something you observe by going on trains

and what not, you can actually show that this

is the reason everything looks the same.

In other words, if the train was at rest on the

platform and you and I were comparing notes and we both find

F = ma, I go to sleep and I'm waking up

and the train is going at a constant speed,

if you can look through the window and look at the objects

in my train, you will say they obey F = ma because

nothing has happened to you. But you will predict I will

also say F = ma because if you see an acceleration,

I will see the same acceleration.

If you see a distance between two masses to be one meter,

I'll also think it's one meter. If the force is 1 over the

square of the distance, we'll agree on the force,

we'll agree on the acceleration,

we'll agree on everything. And once you've proven F =

ma is valid, it follows that every

mechanical phenomenon will behave the same way.

That's the reason things behave the same way.

Yes? Student: If for some

weird reason, suppose different frames of

reference, the rule F = ma was to fail,

what would happen? Professor Ramamurti

Shankar: You mean if the rule failed in the other frame?

Student: Hypothetically. Professor Ramamurti

Shankar: Yes. Suppose, hypothetically,

that happened. Then, it would mean that when

you wake up in the train, you will look at the world

around you, it will look different because F ≠

ma. You will conclude,

when I went to sleep it was F = ma;

when I got up, F ≠ ma,

the train is moving. So, you will have to conclude

that uniform velocity makes detectable changes.

And if you look outside the train, to the other train,

the other train's going backwards.

You can now no longer say, "You're going the other way,

I'm not moving", because the other person will

say, "Hey, F = ma works for me."

It doesn't work for you. So, you're the guy who's

moving." So, you've lost the equal

status with other inertial observers because those for whom

F = ma worked will say they're not moving and for you,

it doesn't work, so you will have to concede you

are moving. So, uniform velocity,

if it makes perceptible changes, can no longer be

considered as relative. It's absolute and if you and I

find each other moving, there may be a real sense in

which I am at rest and you are moving,

because for me F = ma works and for you it doesn't.

Well, that's not what happens. In real life,

you find it works for both of these and either of us can

maintain we are not moving. So now, you've got to fast

forward to about 300 years. This goes on,

no problem with this principle of relativity and 300 years

later, people have discovered

electricity and magnetism and electromagnetism and

electromagnetic waves, which they identify as light.

And then, it was discovered that what you and I call light

is just electric and magnetic fields traveling in space.

You don't have to know what electro-magnetic fields are

right now. They are some measurable

phenomenon. They are like waves.

And the waves have a certain velocity that Maxwell calculated

and that velocity is this famous number 3 times 10 to the eight

meters per second. And the question was,

"For whom is this the velocity?"

For example, you can do a calculation of

waves on a string, something we'll be able to do

in our course. Waves on a string will be some

answer that depends on the tension on the string and the

mass density of the string and that's the velocity as seen by a

person for whom the string is at rest.

But if you calculate the waves of sound in this room -- I talk

to you, you hear me slightly later -- the time it takes to

travel is the velocity of sound in this room.

That is calculated with respect to the air in this room because

the waves travel in the air. In fact, the fact that all of

us are sitting on the planet, which itself is moving at

whatever, 1,100 miles per second,

doesn't matter, because the air is being

carried along so even if the Earth came to a sudden halt,

as far as the velocity of sound in this room is concerned,

it won't matter, because we are carrying the

medium with us. So, people wanted to know what

is the medium which carries the waves of light,

electromagnetic waves? First of all,

the medium is everywhere because--How do we know it's

everywhere? Can anybody tell me?

Yes? Student: It travels

through the vacuum of space. Professor Ramamurti

Shankar: Right. It travels in the vacuum of

space. We can see the Sun;

we can see the stars, so we know the medium is

everywhere. Then, you can sort of ask,

"How dense is the medium?" It turns out that the denser

the medium, the more rapidly signals travel,

in most of the things that we know.

For example, when we look at waves in sound

and when we look at sound waves in a solid,

or in iron, you find in a very dense material,

that the velocity is very high. S,o this medium,

which is called "ether," would have to be very,

very dense to support waves of this incredible velocity.

But then, planets have been moving through this medium for

years and years and not slowing down.

It's a very peculiar medium. But it has to be everywhere so

we are all immersed in this medium because we are able to

send light signals to different parts of the universe.

And the question is, "How fast is the Earth moving

relative to this medium?" You understand?

This medium is all pervasive. We know that we can see the

stars so it's going all the way up to the stars and beyond.

And we are immersed in this and we are drifting around in space.

What is our speed relative to the medium?

That's the question that was asked.

Well, to find the speed relative to the medium,

you calculate the velocity in the medium by Maxwell's theory.

So, here's the medium in which waves travel at a certain speed.

This is planet Earth going around the Sun.

At some instant, you will have a certain

velocity with respect to ether. And therefore,

the velocity of light as seen by you will be modified from

c to c - V. In particular,

suppose the waves are traveling to the right in ether.

Let me draw it this way. The Earth is going at this

instant at the speed V. We expect the speed to be c

- V because part of the speed is neutralized because you

are going along with the waves. You'll see a slower velocity.

So, Mr. Michelson and his assistant

Morley--they did the experiment. And they got the answer equal

to c. What does that mean?

Student: The speed of light [inaudible]

Professor Ramamurti Shankar: No,

no, but you cannot jump to that right now.

If you are following Newtonian physics, your expectation is,

it should be c - V. Yes?

Student: It means that there is no ether.

Professor Ramamurti Shankar: Well,

that's--not so fast, but it certainly means the

following. Well, there's a simpler answer

than that. Yes?

Student: It means that the Earth is moving with respect

to the ether. Professor Ramamurti

Shankar: At what speed? Student: Zero.

Professor Ramamurti Shankar: Zero.

Because you don't have to--Look, you guys are ready to

overthrow everything because you know the answer.

But you've got to put yourself in the place of somebody in the

early 1900s. There's no reason to overthrow

anything. The answer is,

you're going at the speed zero. Of course, you realize,

that it is incredibly fortunate that on the one day Michelson

wants to do the experiment, we happen to be at rest with

respect to the ether. Fine.

But we know that luck is not going to last forever because

you are going around the Sun. On a particular day may be.

But that velocity was such that on that day the Earth was at

rest with respect to the ether. It's clear that six months

later, when we are going the other direction,

you cannot also be at rest with respect to the ether.

But that's what you find when you do the experiment.

You find every day you get the same answer and you jolly well

know you are not at rest. You are moving around the Sun

for sure. Yes?

Student: Did they postulate a drag for the Earth

turning [inaudible] Professor Ramamurti

Shankar: Yes. So, people tried other

solutions. But it is simply a fact that

when you move one way or six months later in the opposite

way, you get the same answer c.

So, one possibility is, you don't want--Look,

don't be ready to do revolutions, try to avoid it.

So, one answer is, look at the speed of sound.

You and I talked to each other then and we talked to each other

six months from now; we get the same speed of sound.

The speed of sound is published in textbooks,

right? Seven hundred and something

miles per hour. How come that doesn't change

from day to day? Anybody here on this side can

tell me why the speed of sound doesn't change from day to day

even though we are moving? No one here can guess?

Student: [inaudible] Professor Ramamurti

Shankar: No, we are moving.

Even six months from now, we get the same speed of sound

in this room. When I talk to you,

does it matter what time of year it is?

Student: The medium is moving along with us?

Professor Ramamurti Shankar: Yes,

we are carrying air. As the Earth moves through

space, it carries the air with you and the speed of the wave is

with respect to the medium. If you can carry the medium

with you, then it doesn't matter how fast you're moving.

So, they tried that. They tried to argue that the

Earth carries ether with it the way it carries air with it.

Then, it's not an accident you are at rest with respect to the

ether because you're taking it with you.

But it's very easy to show by looking at distant stars that

you cannot be doing that. I don't have time to tell you

why that is true. So, you cannot take the ether

with you and you cannot leave it behind, and that's the impasse

people were in. So, it's as if there's a car

that's going to the right at a certain speed c.

You move to the right at some speed, maybe c/2.

I expect you to get a speed c/2.

But you keep getting c. You go three fourths of the

velocity of light; you still get the velocity of

light. That is very contrary to what

we believe. In fact, that's in violent

opposition to this law here. If this V were not a

bullet but a light beam, suppose for me traveling at a

speed c and you're traveling to the right at speed

u, you should get c - u.

That's the inevitable consequence of Newtonian

physics. And you don't get that.

And that was a big problem. So, people tried to fix it up

by doing different models of ether, none of which worked.

And nobody knew why light is behaving in this peculiar

fashion, so that's when Einstein came in and said,

"I know why light is behaving in this peculiar fashion.

It is behaving in this way because if it didn't behave in

this way, if the speed of light depended on how fast I'm moving,

then when I wake up in this train, all I have to do is

measure the speed of light. Originally, I got some number;

now I get to get a different number and the difference will

give me the speed of the train. So, it would have been possible

to detect the velocity of the train without looking outside,

just by doing an experiment with light.

So, even though mechanical laws involving F = ma are the

same, laws of electricity and magnetism would be such that

somehow they would betray your velocity.

And that would mean uniform velocity does make an observable

change because it changes the velocity of light that you would

measure. " But conversely,

the fact that you keep getting the same answer means that

electric and magnetic phenomena are part of the conspiracy to

hide your velocity. Just like mechanical phenomena

won't tell you how fast you're moving, neither will

electromagnetic phenomena. Because to Einstein,

it was very obvious that nature would not design a system in

which mechanical laws are the same but laws of electricity are

different. So, he postulated that all

phenomena, whatever be their nature, will be unaffected by

going to a frame at constant velocity relative to the initial

one. That's a very brave postulate

because it even applies to biological phenomena about which

I'm sure Einstein knew very little.

But he believed that natural phenomena will just follow

either the principle of relativity or they won't.

And that is something you should think about.

Because that was the only reason he had.

He just said, "I don't believe chapters 1

through 10 in our book obey relativity and chapters 20

through 30 where we do E&M doesn't."

These are all natural phenomena that will obey the same

principle, which says all observers that are uniform

relative to motion are equivalent.

Now, that's really based on a lot of faith and even though

scientists generally are opposed to intelligent design,

we all have some bias about the way natural laws were designed;

there's no question about it. You can talk to any practicing

physicist. We have a faith that underlying

laws of nature will have a certain elegance and a certain

beauty and a certain uniformity across all of natural phenomena.

That is a faith that we have. It's not a religious issue;

otherwise, I wouldn't bring it up in the classroom,

but it is certainly the credo of all scientists,

at least all physicists, that there is some elegance in

the laws of nature and we put a lot of money on that faith,

that the laws of nature will do this and will not do this.

Who are we to say that? Who are we to say nature

wouldn't have a system in which mechanics obeys the laws but

electricity and magnetism doesn't?

We haven't run into somebody called nature.

We don't worship a certain deity called nature but we

believe the laws of nature obey that.

So, even though scientists or physicists in particular may not

believe in design by any personal God,

they do believe in this underlying, rational system that

we are trying to uncover. You could be disproven,

you could be wrong in making the assumption,

but here it was right. It was really driven by this

notion that all laws of physics should obey the same principle

of relativity. So, Einstein's postulates are

that light behaves in this way because if it didn't behave in

this way, it would violate the principle

of relativity, whereas we know mechanical

phenomena do and electrical phenomena would not and that

cannot be the case. You have a question?

Student: Why would that not apply to the speed of sound?

Professor Ramamurti Shankar: Yes.

Because in the case of the speed of sound,

you can take the medium with you;

there is no such experiment you could perform.

See, in the train, if you could carry the ether

with you, there's no surprise you would get the same answer.

But we know we cannot carry the medium with us,

that comes from extra-terrestrial experiments.

That's why the velocity of sound is not elevated to a

fundamental velocity on which everybody will agree.

So, the two great postulates--You've got to know

where the postulates come from. Postulate Number 1 is simply a

restatement of the relativity principle.

I'll just say it in one sentence.

Exact wording is not important. All inertial observers are

equivalent.

"Equivalent" means each one of them is as privileged as any

other one to discover the laws of nature.

The laws of nature, we found, are not an accident

related to our state of motion. If I find some laws,

and you're moving relative to me, you'll find the same laws.

And if you and I find each other in relative motion,

you have as much right to claim you are at rest and I am moving

and I have as much right to claim that I am at rest and you

are moving. There is complete symmetry

between observers in uniform relative motion.

There is no symmetry between people in non-uniform [relative]

motion. As I said, non-uniform motion

creates effects which can destroy me and not destroy you.

So, no one's trying to talk their way out of acceleration,

whereas you can talk your way out of uniform velocity.

That's the first principle. So this was there even from the

time of Newton. What is true here is that all

inertial observers are equivalent with respect to all

natural phenomena, meaning all natural laws.

And that is a generalization, when we say "all" instead of

just mechanical.

And the Second Postulate, you call it a postulate because

there is just no way to deduce this,

is that the velocity of light is independent of the state of

motion of the source, of the observer, of everything.

If a light beam is emitted by a moving rocket,

it doesn't matter. If a light beam is seen in a

moving rocket, it doesn't matter.

All people will get the same answer for the velocity of

light. Student: Is there a

reason why the speed of light is constant?

Professor Ramamurti Shankar: No.

That's why it is a postulate. You can show a few things later

on. You can show that if there is

any other speed, which is the same for

everybody, that would have to be the speed of light.

In the final theory of relativity, there are not two or

three velocities that come out the same for everybody.

There is only one velocity that can have the same answer for all

people. That velocity is the velocity

of light. By that I don't mean it has to

be light itself. For example,

gravitational waves travel at the speed of light.

It's not just the light. It has to do with the velocity

of light being a single number which has to have the same value

for everybody. Okay, so it looks like he has

solved a big problem because he has said why light behaves this

way; light behaves this way because

it is part of the big conspiracy to hide uniform motion.

But you will see that you have made a terrible bargain because

once you take these two postulates,

you have restored the relativity principle to all

phenomena. Okay.

You've gone beyond mechanical phenomena to electro-magnetic

phenomena. But you will find that you have

to give up all the other cherished notions of Newtonian

physics. Think about why.

We are saying, here is a car going at 200

miles an hour, according to me.

You get into your own car and follow that car at 50 miles an

hour. You should get 150 but you keep

getting 200. This may not be true for cars

going at the speeds I mentioned but when finally you are talking

about a pulse of light, that is true.

And you've got to agree that is really not compatible with our

daily notions or with the formula I wrote down,

w = v - u. When you put v = c,

w has got to come out to be c.

And that's not a property of the Newtonian transformation.

So, what we are looking for is a new rule for connecting

x and t and x prime and t

prime, such that when the velocities

are computed and applied to the velocity of light,

you get the same answer. That's what we want to do now.

So, here is how we are going to do this.

Now, let's think about it. Let me send a pulse to the

right at speed c. You are going to the right at

three-fourths of c. My Newtonian expectation is,

you should get the speed of the pulse to be one-fourth of

c. But you insist it is c.

So, what will I say to you? What will I accuse you of doing?

I say you should get c/4 and you're getting c.

And you're finding velocity by finding the distance it travels

and dividing by the time so you are jacking up a number like

one-fourth c to c itself.

So, what could make you do that? Yep?

Student: If I perceive that you're moving forward,

or you're contracting your length,

then you're going to measure velocity keeping in mind you

should have had a greater length to begin with.

Professor Ramamurti Shankar: Yes.

So, one option--Let me repeat what he said.

I will say your meter sticks are being somehow shorter.

When you and I were buddies and were on the same train,

we agreed on the meter stick. But now we have gone on the

moving train. I will say there is something

wrong with your meter stick. Not only something wrong.

Specifically, I would say your meter sticks

have shrunk. For example,

if they have shrunk to one-fourth their size,

it is very clear that you would get a velocity of four times

what I expect. But there's another possibility.

Student: Time may be running slow.

Professor Ramamurti Shankar: Your clocks may be

running slow. So, you let the light travel

for four seconds and you thought it was only one second.

That's why you got four times the answer;

or it could be both. But something has to give.

And that is why it is an amazing theory.

That's why it is also amazing to me that somebody who was 26

years old would simply follow the consequence of this theory

and take it wherever it takes you but it is at the very

foundations of space and time that you have to modify.

So, even though you restored the relativity principle and

brought it back to the front, the price you have to pay is to

give up your notions that length is length and time is time.

We used to think a meter stick is a meter stick and a clock is

a clock. If I have a clock that ticks

out one second, you take it on a train,

I expect it to be ticking one second, but we're saying it's

not. So, something has to give in

length measurement or time measurement or both.

And that's what we're going to find out.

So, here's how you find that out.

Let us say that--Maybe I'll do it all on one blackboard because

this is the key to the whole calculation.

You remember now, if there is an event here,

you call it x prime and I call it x.

And according to me, you have traveled the distance

ut.

So, x prime equals x - ut is what we used to say

in the old days. And the converse of that is

x = x prime + ut. But now, we'll admit the fact

that maybe t and t prime are no longer the same.

But that's not all we will do. I will say, whenever you give

me a length x prime, I just don't buy it.

I take any length you give me and I jack it up by a factor of

γ [correction: should have said 1/γ]

to get the length according to me.

And you will take any answer I give you and you will multiply

it by the same factor. Don't worry about this yet!]

In other words, we don't buy our units of

length, so if you say it should be

x prime + ut prime, that's your formula

backwards for me. The coordinate of the event,

according to you, in the old days,

was x prime + ut. Now, we admit that t and

t prime may not be the same.

Then we say, but I will not take your

lengths, I will multiply them by γ to get the lengths

according to me. And you will not take my

expectations, but you will multiply it by the

same γ; that is the symmetry between

the two observers. In other words,

if I think your meter sticks are short and γ

[correction: should have said 1/γ]

is a number less than 1, I'm allowing you to accuse me

of having meter sticks which are short.

It is very interesting. If I said your meter sticks are

short and you say my meter sticks are longer than average,

that's an absolute difference. But we both accuse each other

of using shortened meter sticks, and so we use the same factor

of γ. We are going to find this

γ now. Student: How do you know

that both the distance and time are different?

Professor Ramamurti Shankar: We'll give it the

possibility that they are different and then we will see

that they are different. We know something's wrong with

space and something's wrong with time.

So, we'll not assume that t prime is equal to

t. You have to open up the

possibility. In the end, it may be that the

nature will say t prime is t and something

happens to length alone. But we'll find the answer is

more symmetric. Student: Why do we say

that the symmetry is the same? Professor Ramamurti

Shankar: Because that is the symmetry between the two

observers. If I want to say your meter

sticks are short, why should you concede that?

You should be able to accuse me of saying--The only difference

between you and me is you are moving to the right and I'm

moving to the left. Other than the sign of the

velocity, each person says the other person is moving and so we

will say that any length you give I'll discount by a factor

of γ. By symmetry,

anything I call a length, you will discount by the same

factor of γ. Now, let's apply this.

(x,t) was a certain event,

right? Let's apply it to the following

event. You have to follow this very

carefully. When you and I crossed,

remember that was at the origin of coordinates,

x equals to t equals to zero and x

prime equals to t prime equals to zero.

At that instant when our origins touched,

let us emit a flash of light. Okay, maybe when the origins

touched, there's a spark, the light signal goes out and

the light signal is detected here, by a light detector.

That second event, detection of the pulse,

it has a coordinates (x,t) for me;

it has a coordinates (x prime, t prime) for you.

The same event is given different coordinates.

We've already used the fact that coordinates will be

different but we are saying not only will x prime not be

equal to x; t prime may not be equal

to t either.

Okay, but now let's write down one important condition.

What is the relation between x prime and t

prime in this particular pair of events?

What is x prime? It is the location of the light

pulse after certain time? So, what's the relation of the

location of the detection of the light pulse and the time

t prime? Yes?

Student: [inaudible] Professor Ramamurti

Shankar: He is saying x prime--Do you guys

understand that? Do you agree with that

statement? This is not a random event.

The second event was the detection of a light pulse.

Light pulse left the origin t prime seconds earlier

and has come to this point, according to this guy,

and the ratio of the distance to the time is the velocity of

light. But it is also true that for

me, the light went to distance x in a time t.

So, that's the relation of x to t for me

also. I'm going to use those two

results and combine it with this to find this factor γ

and we will do that now. I want you to multiply the

left-hand side by the left-hand side and the right-hand side by

the right-hand side of that equation.

I hope you understand that in the Galilean days,

in the old days--Let's see what you will say.

I will say, x prime is x - ut because the

origins have shifted by an amount ut.

And you will say x is x prime plus ut

with the same time. Now, I'm saying time is

different. Not only that,

I don't buy your length. If you expect me to have this

length, I say "no." You exaggerated everything.

I'll scale it down by γ and vice versa.

So, if you multiply the left-hand side by the left-hand

side, you get xx prime, the right-hand side you get

γ^(2) times xx prime plus uxt prime

minus ux prime t minus u^(2)tt prime.

Now, divide everything by xx prime.

Then, you get 1 = γ^(2) times 1 + u.

If you divide this by xx prime, you'll get t prime

over x prime. If you divide this by xx

prime, you'll get t/x. And here you get u^(2)

times t/x, times t prime over

x prime.

So, what does that mean? Well, t prime over

x prime is 1/c and t/x is also

1/c because of what I wrote here.

If you want, let me write this as t

prime over x prime equals the c and t over

x equals the c. So, they will cancel.

And I'll get 1 = γ^(2) times 1 - u^(2) over

c^(2) because this t/x is a 1/c and

t prime over x prime is another 1/c.

So, that gives us the result that γ is 1 over square

root of 1 - u^(2) over c^(2).

If you plug the γ back in, you will find x prime

is x - ut divided by [square root of]

1 - u^(2) over c^(2).

Now, once you do that, once you got the relation

between x prime and x, you can go to the

lower equation and solve for t prime.

I don't feel like doing that. It's a simple algebraic

equation once you've got x prime how to solve for

t prime. Take the second of the two

equations and solve for t prime.

That detail I won't fill out, but you will get t prime

is t - ux over c^(2) divided by this.

So, I've not done every step but I've given you all the

things you need to do the one step.

There are equations up there that relate x and

x prime to t and t prime so if you can get

x prime in terms of x and t,

the other equation that you solve will give you t

prime in terms of x and t.

And this is the result.

Okay. This guy deserves two boxes

because it is the greatest result from relativity;

it's called the Lorentz transformation.

And we've been able to derive the Lorentz transformation with

what little we know. And you can see,

you can be a kid in high school and you can do this.

There's no calculus or anything else involved,

other than being open to the fact that the velocity of light

behaves in this strange way. Yes?

Student: How do you get t/x = c?

Professor Ramamurti Shankar: Why is t/x =

c? Oh, of course,

you're quite right. So, you caught the mistake here.

t/x is 1 over c and 1/c.

I really meant to write x = ct.

Yes? Student: [inaudible]

Professor Ramamurti Shankar: No.

If you define γ to be the absolute value by which you

transfer lengths from you to me, then you can take the positive

root. Student: [inaudible]

Professor Ramamurti Shankar: Well,

I can also tell you other reasons.

Let's take this formula here. Let's take the case where the

velocity is very small compared to the velocity of light.

That u/c is a very small number.

This number is almost 1 and I get x prime because x

- ut, which I know to be the correct answer at lower

velocities. If you pick the minus sign,

I'll get minus of x - ut, that's not the right

answer, not even close to the right answer.

At low velocities, if you go to velocity u

over c much less than 1, you have got to get back to

Galilean transformation. You can see if u over

c goes to zero. You can forget all about this

factor here. You get x prime is

x - ut and here, u over c can be

neglected, forget all that,

you get t prime equals t.

So, this coordinate transformation would reduce to

the Galilean transformation if the velocity between me and you

is much smaller than the velocity of light.

So, the formula really kicks in for velocities comparable to

velocity of light. Yes?

Student: [inaudible] What happens if u >

c? Professor Ramamurti

Shankar: Well, you start getting crazy

answers, right? You can already see that the

theory will not admit velocities bigger than the velocity of

light. You can already see it in this

formula. That tells you that the one

single velocity that you wanted to be the same for everybody is

also the greatest possible velocity;

that no observer can move at this speed with respect to

another observer that is equal to what is in excess of the

speed of light. So, the speed of light,

which came out to be a constant in the beginning of the theory,

is also turning out to be the upper limit on possible

velocities. That's the origin of the

statement that no observer can travel at a speed bigger than

light, but we'll discuss it more and more.

But you have to understand what it is that is being derived,

what is the meaning of this formula here.

What is it telling you? If I say, this is called the

Lorenz transformations, what do they tell you?

What are these numbers and what's their significance?

Would you like to try? Student: Well,

one thing that they tell you is if u happens to be

greater than c [inaudible]

Professor Ramamurti Shankar: No,

no, I don't mean what happens in the formula in special cases.

What is it relating? What is xt and what is

x prime t prime? Student:

Okay, so x prime would be the distance that the person

who's traveling at the higher speed experiences.

And [inaudible] and so for that person,

distance is going to seem shorter?

Professor Ramamurti Shankar: No.

See, I'm not even telling you to get consequences of the

equation. What are the numbers x

and t in this equation? And what are the numbers

x prime and t prime in this equation?

Student: x and t are distance and time

for a person who is in an inertial frame of reference,

who is not moving. Professor Ramamurti

Shankar: Right. Student: And x

prime and t prime are distance and time for the person

who is moving at the speed of [inaudible]

Professor Ramamurti Shankar: And when you say

distance and time, what do you mean,

distance and time? Student: The way that

the length of the distance will seem to them that they travel.

Professor Ramamurti Shankar: But what is

happening at xt; it's the coordinates of what?

Student: Of their location [inaudible]

Professor Ramamurti Shankar: They are located at

zero, zero. Right?

What's happening at x and t?

Yes? Student: It's observing

the event. Professor Ramamurti

Shankar: It's the event. The key I was looking for is an

event. You've got to understand what

the formula is connecting. Things are happening in space

and in time, right? Something happens here.

That something has a spatial coordinate and a time

coordinate, according to two observers.

The observers originally had their origins and their clocks

coincide when they passed; that's how they're related.

And one is moving to the right at speed u.

Then, the claim is that if one event had a coordinate x

and t for one person, for the other person moving to

the right at speed u, the same event would have

coordinates x prime and t prime and the relation

between x and t and x prime and t

prime is this. Yes?

Student: The two observers [inaudible]

don't they observe different laws and [inaudible]

Professor Ramamurti Shankar: No.

The fact that an event has different coordinates doesn't

mean that you are observing different laws.

For example, let's take that fire

extinguisher. We look at it,

it's obeying F = ma, right?

The coordinates of the fire extinguisher with me as the

origin is quite different from you as the origin.

Student: [inaudible] Professor Ramamurti

Shankar: You mean in these new equations?

Yes, in these new equations, F = ma will not work;

that's correct. Student: They are

inertial references, even though they are moving?

Professor Ramamurti Shankar: Ah.

Yes. So, the point is the laws of

Newton themselves have to be modified.

F = ma will be modified in a certain way but the new

modified laws will reduce to F = ma at low velocities,

which is why in the old days it looked like F = ma.

But there will be new laws, but they will also have the

property that when I measure them I'll get the laws that will

agree with what you measure. Yes?

Student: Does individual values for time or distance

still have to agree? Professor Ramamurti

Shankar: The coordinates of an event will differ from person

to person. That's not the same as saying

the laws as deduced will be different.

For example, there are two stars which are

attracted to each other by gravitation and they are

orbiting around their common center of mass.

If I see them, I will find that they obey the

law of gravitation with m_1m

_2 over r^(2) where r is

the distance between the points and the acceleration is whatever

I think it is. You can go on a rocket and look

at the same two stars. They will be in a more

complicated motion, maybe the whole system will be

drifting a little to you, but their acceleration will be

the same as what I get and the force between them will also be

the same as what I get and the laws that you would deduce by

looking at that star would be the same laws that I would

deduce. So, that's a difference between

the laws being the same and the coordinates being the same.

No one said x prime and x are the same in that

equation there. They are different.

We are looking at it from different vantage points.

But the fact is that force is equal to mass times acceleration

is the same for the two people. Okay.

The laws will be the same but things won't look the same.

For example, you can stand on your head,

don't even have to go to another frame of reference;

you can stand on your head, your z coordinate is the

minus of my z coordinate; to every point I give a

z, you will give a minus z.

But the world, even though you are a little

messed up and want to stand on your head,

you have every right to do that and you will find that F =

ma. So the point is,

the way we see events may depend on their origin or

coordinates, but the laws we deduce are to

be distinguished from the perception that we have.

Okay? For example,

if I'm on the ground, I send a piece of chalk,

it goes up and it goes down. If you see me from a moving

train, you would think it went on a parabola.

So, no one says the chalk will go up and down for you.

For you, it will go like this but its motion will still obey

F = ma, is what I'm saying.

That's all you really mean by saying things look the same.

So, what you have to understand is that Lorentz transformations

are the way to relate a pair of events, given events.

Here's a simple example. If you live in the xy

plane, there's a point here. It's not an event,

it's simply a point in the xy plane.

You measure it this way and that way and you call it the

coordinates. If somebody else picks a

different coordinate system with an angle θ,

that person will say that's x prime and that is

y prime and x prime and y prime are not

the same. I remind you,

x prime is x cos> θ [delete "minus

y"] plus y sin θ,

etc., and y prime is something else.

θ is the angle between the two observers.

So, the point is the point. It certainly looks different to

the two people, but the same point has two

coordinates. Similarly, the same event,

like the collision of two cars, will have different events for

different people. That's not the new part.

The new part is that the rules for connecting xt to

x't', is quite different from the Galilean rules,

new rules. It's what you guys have to

understand. And finally,

why did Einstein get the credit for turning the world into four

dimensions instead of three? After all, x and

t were present there, too.

The point is, t prime is always equal

to t no matter how you move,

whereas in the Einstein theory, x and t get

scrambled into x prime and t prime just the way

x and y get scrambled into each other when

you rotate your axis. So, to have time as another

variable that doesn't transform at all is not the same as making

it into a coordinate. The four-dimensional world of

Einstein is four-dimensional because space and time mix with

each other when you change your frame of reference.

That's what makes t now a coordinate as previously it

was something the same for all people.

let's begin now. First of all,

I'm assuming all of you have some idea what special

relativity means. There are two theories of

relativity, one is the special theory and one is the general

theory. The general theory is something

that we won't do in any detail. Special theory is something we

will do in reasonable detail. So, it's good to begin by

asking some of you what is your present understanding of what

the subject is all about. Yes, sir?

The Yale cap, what do you think it's about?

Student: It's about relative speed in two reference

systems. Professor Ramamurti

Shankar: Okay, it's about relative speed in

two reference systems. I'll come to you;

then I'll come to you. Student: It's based on

the postulate that the laws of physics are the same in any two

references moving in uniform motion relative to one another

and the speed of light is constant in all references.

Professor Ramamurti Shankar: Okay,

I will take the last row there. Student: [inaudible]

Professor Ramamurti Shankar: Okay,

so what I've heard so far is that the laws of physics are the

same for two people who are both in inertial frames of reference

and the velocity of light's a constant.

Right. That's certainly the way we

understand the special relativity theory.

But it's a very old one. It's been going on long before

Einstein came. There was a theory of

relativity at the time of Newton and that's where I want to

begin. Relativity is not a new idea at

all, it's an old one. And the old idea can be

illustrated in this way and it will agree with your own

experience. So, the standard technique for

all of relativity is to get these high speed trains.

I'm going to have our own high speed train;

this is the top view of the train.

And like in everything I do, we'll get away with the lowest

number of dimensions, which happens to be this one

spatial dimension and of course there is time.

So, the train is moving along the x axis.

You are in this train. You board the train and all the

blinds are closed because you don't want to look outside.

That's not because you're traveling through some parts of

New Jersey; you don't want to look outside

for this particular experiment. You get into the train,

you settle down and you explore the world around you.

You pour yourself a drink, you play pool,

you juggle some ping pong balls, tennis balls,

and you have a certain awareness of what's happening,

namely, your understanding of the mechanical world,

and then you go to sleep. When you are sleeping,

some unseen hand gives to this train a large velocity,

200 miles an hour. The question is,

"When you wake up, can you tell if you're moving

or not?" That's the whole question.

Will this speed, whatever I gave you,

200 miles per hour--will it do anything to you in this train

that will betray that velocity? So, when you wake up will you

say I'm moving or not? Now, you might say,

I'm not moving because I'm on Amtrak and I know this train is

not going anywhere. That kind of sociological

reason, by the way, there are many of them,

you cannot invoke. You can only say,

"I'm on this train. Is anything different?"

And the claim is that nothing will be different.

You just will not know you are moving.

Now, if the train picks up speed, or slows down,

you will know right away. If it picks up speed or

accelerates, you find yourself pushed against the back of the

seat or if the driver slams on the brake,

you will slam into the front of the seat in front of you.

No one is saying that when the motion is accelerated,

you will not know. Accelerated motion can be

detected in a closed train without looking outside.

The question is uniform velocity, no matter how high,

can that be perceived? Can that be detected?

So, at the time of Galileo and Newton everybody agreed that you

cannot detect it. Remember that if you started

out and Newton's laws worked for you, you are called an inertial

observer. One of the laws you want is,

if you leave something, it should stay where it is.

When the train is accelerating, that won't be true.

You leave things on the floor when it's accelerating,

things will slide backwards. So, with no apparent force

acting on it, things will begin to

accelerate; that's a non-inertial frame.

We are not interested in that. You started out as an observer

for whom the laws of Newton work, the laws of inertia work,

F = ma, then you go to sleep and you wake up.

So, when I said everything looks the same,

I really meant that the laws of Newton continue to be the same

because if the laws of Newton are the same,

everything will look the same. That's what it means to say

"everything looks the same." Our expectations of what

happens when I throw it up or what happens when two billiard

balls collide, everything is connected to the

laws of Newton. So, the claim is,

the laws of Newton will be unchanged when this velocity is

added on to you. Now, we should be clear about

one thing. If there is a train next to you

in the beginning -- let me just put it on this side for

convenience -- and you got in and you boarded this train but

you looked at this train and it was not moving.

If you lift the blind and look through, you'll see the other

train and there's another passenger in the other train and

you look at each other, you're not moving.

When you wake up after this brief nap, you find when you

look outside the other train is moving at 200 miles an hour.

The question is, "Can you tell if it's you who's

responsible for this relative motion,

or maybe nothing happened to you and the other train is

moving the opposite way?" And the claim of relativity is

that you really cannot tell. You can tell there is motion

between the two trains that wasn't there before.

That's very clear if you look outside but there is no way to

tell what actually happened when you were sleeping.

Whether you were given the velocity of 200 to the right or

the other train was given a velocity of 200 to the left or

maybe a combination of the two, you just cannot tell.

That's the word "relative." So far -- I didn't tell you --

if you have only one train, what I told you earlier,

is that uniform velocity does not leave its imprint on

anything you can measure. If you look outside,

of course you can see the motion of the other train,

but you still cannot tell who is moving.

You cannot distinguish between different possibilities.

So, you have every right to insist that you are not moving

and the other train is moving the opposite way.

Once again, you can make this argument only for uniform

relative motion. If your train is accelerating,

now I'm saying it as if it is an absolute thing,

and it is. You cannot say,

"I'm not accelerating, the other train is accelerating

in the other direction." You cannot say that because

you're the one who is barfing up and throwing up and slamming

your head on the wall; nothing is happening to the

other person. You cannot say "I'm still in

the same frame, you are going the opposite

way." If you are going the opposite

way, why am I throwing up? Or if you are in a rocket and

the rocket's taking off and the G forces are enormous,

many times your weight, it is the astronauts who are

going through the discomfort. At that time they cannot say we

are at rest and everyone is going the opposite way because

no one else is in danger, but they are.

So, accelerated motion will produce effects.

You cannot talk your way out of that.

But uniform velocity will produce no effects on you and no

effects on the other person. You can detect relative motion

but you cannot in any sense maintain that you are moving and

he's not or that he's moving and you are not.

You can say, "I am at rest,

things are the same as before, the train is moving the

opposite way." Now, if you go in the Amtrak

and you look outside and you don't see another train,

but you see the landscape, you see trees and cows and

everything, going at 200 miles an hour in the opposite

direction, you have some reason to believe

that probably the ground is not moving and you are moving.

But that's just based on what I called earlier some sociological

factors. In other words,

it's completely possible to devise an experiment in which

somebody puts the whole landscape on wheels and when you

go to sleep the landscapes, cows and trees are made to move

the opposite way. Not very likely,

but that's because we know in practice no one is going to

bother to do that just to fool you.

But if that did happen, you won't know the difference.

So, the reason we rule that out is we know some extraneous

things not connected to the laws of physics.

That's why we don't like to open the window and look at the

landscape because then we have a bias.

Open the window and look at another train and you just won't

know. That is the principle of

relativity, that uniform motion between two observers,

both of whom are inertial, is relative.

Each one can insist that he or she is not moving;

the other person is the one who is moving.

Of course, now, if the two, in reality,

if the two trains were at rest--Let's imagine my train got

accelerated. So, during the time it was

accelerated, I would know, but if I was sleeping at the

time, I don't know and when I wake up

and the acceleration is gone and the velocity is constant,

that's when I say, "I just cannot tell."

All right. Now, let's show once and for

all that the laws of Newton are not going to be modified.

So, you find the laws of Newton before you go to sleep,

you wake up, you find them again,

you'll get the same laws; that's the claim.

I hope you understand that all the mechanical things you see in

the world around you come just from F = ma.

We have seen projectiles and collision of billiard balls and

rockets; they're all Newtonian mechanics.

So, to say that things will look the same is to say the laws

of Newton that you will deduce before and after waking up will

be the same. So, let's show that.

When you show that, you're really done with it once

and for all. So, let's do the following.

Here is the x axis and here is my frame of reference.

This is my x axis. Let's call this the origin.

The frame goes to negative and positive x values.

Pick some object sitting at the point x.

Now, we are going to first define the notion of an event.

An event is something that happens at a certain place at a

certain time; that's called an event.

For example, if there's a little firecracker

going off somewhere at some time, the x is where it

happened and the t is when it happened.

So, this is space-time. Once again, space-time does not

require Einstein coming in at all.

We have known for thousands of years that if you want to set up

a meeting with somebody, you've got to say where and

you've got to say when and things do happen in space-time.

The fact that you need x and t,

or, if you're living in three spatial dimensions,

the fact that you need x, y,

z and t is not new.

That is not the revolution Einstein created.

The fact that you need four coordinates to label an event is

nothing new. What he did that is new will be

clear later. So, does everyone understand

what an event means? Okay?

An event is something that happens and to say exactly where

and when it happened, in our world of one dimension,

we give it an x and we give it a t.

Now, that's me, and I'm going to give my frame

of reference the name S. It turns out S is not

just based on just my name. This is the canonical name for

two observers, one is called S and one

is called S prime. So, S prime,

let's say, is you. So, your frame of reference is

going to be taken to be sliding relative to mine.

So, let's draw a y axis here.

We don't really deal with the y coordinate but just to

give you a feeling, this is my y axis;

that's my x axis; this is my origin.

y is not going to play a big role.

Now, you are sliding this way, to the right,

and your speed or velocity is always denoted by u.

Some number of meters per second, you are zooming to the

right at some speed u. So, imagine now you are going

past me. At some instant--I'm sitting

here at the origin of coordinates, you cross me and

then a little later you are somewhere there.

So, that's your y frame axis;

that's your origin. And the same event you say has

a coordinate x prime.

We arrange it so that when you zoom past me,

you set your clock to zero and I set my clock to zero.

When you want to set the clock to zero is completely arbitrary.

So, we will decide right when you pass me.

I'm at the origin of my coordinates, you're at the

origin of your coordinates. When you pass me,

we'll click our stop watches and we will set the time to

zero. So, here's an event.

You and I crossed. What are the coordinates for

that event? According to me,

that event occurred at x = 0 and the time was chosen to

be zero. According to you,

your origin was also on top of my origin, so x prime was

zero and the time is just the time.

Everybody has a single time, and that time is called zero.

That is one event. We made the coordinate of the

event zero zero for both you and me.

It is zero in space because my origin crossed your origin.

That crossing took place at my origin;

that's why my x is zero and took place at your origin,

that's where your x prime is zero and the common

time we chose to be zero by convention.

Then, we want a second event. So, let's say the second event

is some firecracker going off here.

Here is something I should explain that I used to forget in

the previous years. When I say I am moving,

I imagine I am part of a huge team of people who are all

moving with me. So, I've got agents all over

the x axis who are my eyes and ears;

they are looking out for me. So, even though I am here,

if there's a firecracker exploding here,

my guys will tell me. And you are carrying your own

agents. Let's say at every point

x you have a reporter, x = 1,2,

3,4; there are people sitting and

watching. So, when I say I see something,

I really mean me and my buddies, all traveling the same

train at the same speed, all over space taking notes on

what's happening. We'll pool our information

later but we know this explosion took place here.

I'll simplify it by saying I know an explosion took place

here at location x at time t.

So, this is our crossing. This event is when we crossed.

Then, there is a firecracker. For the firecracker,

I have to give some events [should have said

"coordinates"]. I say it took place at location

x at time t.

What do you say? You measure the distance from

your origin, you call this some x prime,

the time is still t. In Newtonian mechanics,

the time is just the time. How many seconds have passed is

the same for everybody. The question is,

"What is the relation between x prime and x?"

That's what we want to think about.

So, you guys should think before I write down the answer.

What's the relation of x prime to x?

Well, this event took place at time t,

so I know that your origin is to the right by an amount

u times t.

So, the distance from your origin for this event,

I maintain is x prime is x - ut.

Again, I want all of you to follow everything.

These are all simple notions. Our origins coincided at zero

time for the event that occurs at time t.

Therefore, in a sense, you are rushing toward the

event. You've gone a distance

ut. Therefore, the distance from

your origin to the event will be less than mine by this amount

ut. This is the law of

transformation of coordinates in Newtonian mechanics.

If you have an event--if you want, formally,

you can define it time, t prime for the primed

observer. It goes without saying that

t prime and t are the same.

There's no notion of time for me and time for you.

There's universal time in Newtonian mechanics.

It just runs. We can call some time a zero.

Once we have agreed, if you say you and I met at

t = 0 and an explosion took place 5 seconds after our

meeting, it's going to be 5 seconds

after our meeting for me and it's going to be 5 seconds after

the meeting for you. The time difference between two

events is the same for all people.

This is called the Galilean transformation.

What are the consequences of Galilean transformation?

Well, let's look at the fact that x prime is x -

ut.

Remember, everything is varying with time.

So x prime is a function of time and x is a

function of time, if you are watching a moving

particle. Suppose this firecracker is not

just one event, but it's a moving object.

Let's give the object some speed;

it's moving to the right. Then the velocity,

according to me--I'm going to call v as dx/dt is

the velocity. Let's just call it a bullet,

according to S. Then w -- it's the

standard name -- is the velocity of the bullet according to

S prime. So, what I've done is,

I first took one event and I gave it some coordinates and I

told you how to transform the coordinate from one person to

the other person. But now, take that point

x not to be a fixed location but a moving object,

so that as a function of time that body is moving.

Then its velocity at any time is dx prime dt

according to you; that is the dx/dt

according to me, minus the derivative of this,

which is u. Now, does that make sense?

This should agree with common sense.

For example, if that bullet is going at 600

miles per hour to the right, that is 600 for me,

and you are going to the right in your train at 200 miles per

hour, you should measure the bullet speed to be reduced by

200 and you should get 400. That's all it means.

The two people will disagree on the velocity of the bullet

because they are moving relative to each other.

This is the way you will add velocities.

But let's look at the acceleration.

dw/dt is going to be dv/dt - 0 because

u is a constant.

That means you and I agree on the acceleration of the body.

We disagree on where it is. We disagree on how fast the

bullet's moving. But we agree on the

acceleration of the body because all I've done is add a constant

velocity to everything you see. Therefore, if according to you

the velocity of the body is not changing, according to me the

velocity of the body is not changing,

because the constant added will drop out of the difference.

Or, if the body has an acceleration,

we'll both get the same answer for the acceleration.

So, that is the common acceleration a.

So, if you like, a-prime is the same as

a.

So, the acceleration of bodies doesn't change when you go from

one frame of reference to another one going at a constant

speed.

All right, so let's look at F = ma,

which is md^(2)x/dt^(2) is equal to some force on the

body. And you look at the body and

you say d^(2)x prime over dt^(2) is the force on

your body. First, I want to convince you

we wanted to see that the left-hand sides are equal

because the acceleration's the same.

Then, I want to convince you that the right-hand sides are

also going to be equal. I can take many examples but

eventually you will get the point.

Let us not consider one body, but let's consider two bodies.

Two bodies are feeling a certain force due to,

say, gravitation. And gravitation is,

of course, a force in three dimensions but let's write the

force in just one dimension. And let's say the force of

gravity is equal to 1 over x_1 -

x_2. Force on 1 due to 2 and the

force on 2 due to 1 will be minus 1 over x_1 -

x_2. The real forces are separation

in three dimensions but this is a fictitious force.

I want to call it gravity. It is any force that depends on

the coordinates of the two particles.

So, I will say m_1d^(2)x

_1 over dt^(2) is 1 over

x_1 - x_2.

And m_2d^(2)x _2 over

dt^(2) is minus 1 over x_1 minus--I

have forgotten constants like g and

m_1 and m_2.

They don't matter for this purpose.

So, here are two bodies. They feel a force for each

other and I've discovered what the force is.

It's 1 over x_1 - x_2.

I don't care if it's 1 over x_1 -

x_2 or (x_1 -

x_2)^(2); that's not important.

What's important is, it depends on x_1

- x_2. You come along and you study

the same two masses. What will you say is happening?

You will say, m_1d^(2)x

_1 prime over dt^(2) is equal to 1 over

x_1 prime minus x_2 prime.

Maybe I will--I'm sorry. Let me do it a little better.

I can tell you what you will see.

Given this is what I see, I can tell you what you will

see. Let's do that in our head.

We know that the acceleration is the same for any mass so I'm

going to write this thing as m of dx over prime

dt^(2). In other words,

the acceleration according to me is the same as the

acceleration according to you. Then, I'm also going to write

the right-hand side as x_1 prime minus

x_2 prime. Do you understand that?

If there are two bodies feeling a force, if you see it from a

moving train, the distance between the two

bodies is the same for you and me,

because x_1 prime is x_1 -

ut and x_2 prime is x_2 -

ut. Take the difference;

the difference between the location of the particles is the

same for you and me. Acceleration is the same,

mass is postulated to be the same, so I know that you will

get the same law that I get. You will get F = ma;

your acceleration will be the same as mine;

the force you attribute between the two bodies will also be the

same. That is why I know that you

will also deduce the same Newtonian laws that I will.

You can also say it differently. If I woke up from my nap and I

am now in a moving train and I examine the world around me,

I'm going to get the same F = ma.

Because as seen by a person on the ground, the masses obey F

= ma. I am in this moving train now

but I have the same acceleration for each mass and I have the

same force. So, if you want,

I'll complete the second equation.

m_2d^(2) x_2 prime over

dt^(2) will be minus 1 over x_1 prime

minus x_2 prime.

If this is a little difficult, we should talk about this.

I'm telling you that if I deduce F = ma and the

F depends on the separation between the

particles, then I'm sure that you will

find the same laws of motion because the acceleration is the

same that I get because we have seen a-prime is the same

as a. And the force will also be the

same because the force depends on the separation between

particles. And that doesn't depend on

which train you're in or it's not affected by adding a

constant velocity to the frame of reference.

So, if you like, this is the way you prove in

Newtonian mechanics the principle of relativity.

So, not only is it something you observe by going on trains

and what not, you can actually show that this

is the reason everything looks the same.

In other words, if the train was at rest on the

platform and you and I were comparing notes and we both find

F = ma, I go to sleep and I'm waking up

and the train is going at a constant speed,

if you can look through the window and look at the objects

in my train, you will say they obey F = ma because

nothing has happened to you. But you will predict I will

also say F = ma because if you see an acceleration,

I will see the same acceleration.

If you see a distance between two masses to be one meter,

I'll also think it's one meter. If the force is 1 over the

square of the distance, we'll agree on the force,

we'll agree on the acceleration,

we'll agree on everything. And once you've proven F =

ma is valid, it follows that every

mechanical phenomenon will behave the same way.

That's the reason things behave the same way.

Yes? Student: If for some

weird reason, suppose different frames of

reference, the rule F = ma was to fail,

what would happen? Professor Ramamurti

Shankar: You mean if the rule failed in the other frame?

Student: Hypothetically. Professor Ramamurti

Shankar: Yes. Suppose, hypothetically,

that happened. Then, it would mean that when

you wake up in the train, you will look at the world

around you, it will look different because F ≠

ma. You will conclude,

when I went to sleep it was F = ma;

when I got up, F ≠ ma,

the train is moving. So, you will have to conclude

that uniform velocity makes detectable changes.

And if you look outside the train, to the other train,

the other train's going backwards.

You can now no longer say, "You're going the other way,

I'm not moving", because the other person will

say, "Hey, F = ma works for me."

It doesn't work for you. So, you're the guy who's

moving." So, you've lost the equal

status with other inertial observers because those for whom

F = ma worked will say they're not moving and for you,

it doesn't work, so you will have to concede you

are moving. So, uniform velocity,

if it makes perceptible changes, can no longer be

considered as relative. It's absolute and if you and I

find each other moving, there may be a real sense in

which I am at rest and you are moving,

because for me F = ma works and for you it doesn't.

Well, that's not what happens. In real life,

you find it works for both of these and either of us can

maintain we are not moving. So now, you've got to fast

forward to about 300 years. This goes on,

no problem with this principle of relativity and 300 years

later, people have discovered

electricity and magnetism and electromagnetism and

electromagnetic waves, which they identify as light.

And then, it was discovered that what you and I call light

is just electric and magnetic fields traveling in space.

You don't have to know what electro-magnetic fields are

right now. They are some measurable

phenomenon. They are like waves.

And the waves have a certain velocity that Maxwell calculated

and that velocity is this famous number 3 times 10 to the eight

meters per second. And the question was,

"For whom is this the velocity?"

For example, you can do a calculation of

waves on a string, something we'll be able to do

in our course. Waves on a string will be some

answer that depends on the tension on the string and the

mass density of the string and that's the velocity as seen by a

person for whom the string is at rest.

But if you calculate the waves of sound in this room -- I talk

to you, you hear me slightly later -- the time it takes to

travel is the velocity of sound in this room.

That is calculated with respect to the air in this room because

the waves travel in the air. In fact, the fact that all of

us are sitting on the planet, which itself is moving at

whatever, 1,100 miles per second,

doesn't matter, because the air is being

carried along so even if the Earth came to a sudden halt,

as far as the velocity of sound in this room is concerned,

it won't matter, because we are carrying the

medium with us. So, people wanted to know what

is the medium which carries the waves of light,

electromagnetic waves? First of all,

the medium is everywhere because--How do we know it's

everywhere? Can anybody tell me?

Yes? Student: It travels

through the vacuum of space. Professor Ramamurti

Shankar: Right. It travels in the vacuum of

space. We can see the Sun;

we can see the stars, so we know the medium is

everywhere. Then, you can sort of ask,

"How dense is the medium?" It turns out that the denser

the medium, the more rapidly signals travel,

in most of the things that we know.

For example, when we look at waves in sound

and when we look at sound waves in a solid,

or in iron, you find in a very dense material,

that the velocity is very high. S,o this medium,

which is called "ether," would have to be very,

very dense to support waves of this incredible velocity.

But then, planets have been moving through this medium for

years and years and not slowing down.

It's a very peculiar medium. But it has to be everywhere so

we are all immersed in this medium because we are able to

send light signals to different parts of the universe.

And the question is, "How fast is the Earth moving

relative to this medium?" You understand?

This medium is all pervasive. We know that we can see the

stars so it's going all the way up to the stars and beyond.

And we are immersed in this and we are drifting around in space.

What is our speed relative to the medium?

That's the question that was asked.

Well, to find the speed relative to the medium,

you calculate the velocity in the medium by Maxwell's theory.

So, here's the medium in which waves travel at a certain speed.

This is planet Earth going around the Sun.

At some instant, you will have a certain

velocity with respect to ether. And therefore,

the velocity of light as seen by you will be modified from

c to c - V. In particular,

suppose the waves are traveling to the right in ether.

Let me draw it this way. The Earth is going at this

instant at the speed V. We expect the speed to be c

- V because part of the speed is neutralized because you

are going along with the waves. You'll see a slower velocity.

So, Mr. Michelson and his assistant

Morley--they did the experiment. And they got the answer equal

to c. What does that mean?

Student: The speed of light [inaudible]

Professor Ramamurti Shankar: No,

no, but you cannot jump to that right now.

If you are following Newtonian physics, your expectation is,

it should be c - V. Yes?

Student: It means that there is no ether.

Professor Ramamurti Shankar: Well,

that's--not so fast, but it certainly means the

following. Well, there's a simpler answer

than that. Yes?

Student: It means that the Earth is moving with respect

to the ether. Professor Ramamurti

Shankar: At what speed? Student: Zero.

Professor Ramamurti Shankar: Zero.

Because you don't have to--Look, you guys are ready to

overthrow everything because you know the answer.

But you've got to put yourself in the place of somebody in the

early 1900s. There's no reason to overthrow

anything. The answer is,

you're going at the speed zero. Of course, you realize,

that it is incredibly fortunate that on the one day Michelson

wants to do the experiment, we happen to be at rest with

respect to the ether. Fine.

But we know that luck is not going to last forever because

you are going around the Sun. On a particular day may be.

But that velocity was such that on that day the Earth was at

rest with respect to the ether. It's clear that six months

later, when we are going the other direction,

you cannot also be at rest with respect to the ether.

But that's what you find when you do the experiment.

You find every day you get the same answer and you jolly well

know you are not at rest. You are moving around the Sun

for sure. Yes?

Student: Did they postulate a drag for the Earth

turning [inaudible] Professor Ramamurti

Shankar: Yes. So, people tried other

solutions. But it is simply a fact that

when you move one way or six months later in the opposite

way, you get the same answer c.

So, one possibility is, you don't want--Look,

don't be ready to do revolutions, try to avoid it.

So, one answer is, look at the speed of sound.

You and I talked to each other then and we talked to each other

six months from now; we get the same speed of sound.

The speed of sound is published in textbooks,

right? Seven hundred and something

miles per hour. How come that doesn't change

from day to day? Anybody here on this side can

tell me why the speed of sound doesn't change from day to day

even though we are moving? No one here can guess?

Student: [inaudible] Professor Ramamurti

Shankar: No, we are moving.

Even six months from now, we get the same speed of sound

in this room. When I talk to you,

does it matter what time of year it is?

Student: The medium is moving along with us?

Professor Ramamurti Shankar: Yes,

we are carrying air. As the Earth moves through

space, it carries the air with you and the speed of the wave is

with respect to the medium. If you can carry the medium

with you, then it doesn't matter how fast you're moving.

So, they tried that. They tried to argue that the

Earth carries ether with it the way it carries air with it.

Then, it's not an accident you are at rest with respect to the

ether because you're taking it with you.

But it's very easy to show by looking at distant stars that

you cannot be doing that. I don't have time to tell you

why that is true. So, you cannot take the ether

with you and you cannot leave it behind, and that's the impasse

people were in. So, it's as if there's a car

that's going to the right at a certain speed c.

You move to the right at some speed, maybe c/2.

I expect you to get a speed c/2.

But you keep getting c. You go three fourths of the

velocity of light; you still get the velocity of

light. That is very contrary to what

we believe. In fact, that's in violent

opposition to this law here. If this V were not a

bullet but a light beam, suppose for me traveling at a

speed c and you're traveling to the right at speed

u, you should get c - u.

That's the inevitable consequence of Newtonian

physics. And you don't get that.

And that was a big problem. So, people tried to fix it up

by doing different models of ether, none of which worked.

And nobody knew why light is behaving in this peculiar

fashion, so that's when Einstein came in and said,

"I know why light is behaving in this peculiar fashion.

It is behaving in this way because if it didn't behave in

this way, if the speed of light depended on how fast I'm moving,

then when I wake up in this train, all I have to do is

measure the speed of light. Originally, I got some number;

now I get to get a different number and the difference will

give me the speed of the train. So, it would have been possible

to detect the velocity of the train without looking outside,

just by doing an experiment with light.

So, even though mechanical laws involving F = ma are the

same, laws of electricity and magnetism would be such that

somehow they would betray your velocity.

And that would mean uniform velocity does make an observable

change because it changes the velocity of light that you would

measure. " But conversely,

the fact that you keep getting the same answer means that

electric and magnetic phenomena are part of the conspiracy to

hide your velocity. Just like mechanical phenomena

won't tell you how fast you're moving, neither will

electromagnetic phenomena. Because to Einstein,

it was very obvious that nature would not design a system in

which mechanical laws are the same but laws of electricity are

different. So, he postulated that all

phenomena, whatever be their nature, will be unaffected by

going to a frame at constant velocity relative to the initial

one. That's a very brave postulate

because it even applies to biological phenomena about which

I'm sure Einstein knew very little.

But he believed that natural phenomena will just follow

either the principle of relativity or they won't.

And that is something you should think about.

Because that was the only reason he had.

He just said, "I don't believe chapters 1

through 10 in our book obey relativity and chapters 20

through 30 where we do E&M doesn't."

These are all natural phenomena that will obey the same

principle, which says all observers that are uniform

relative to motion are equivalent.

Now, that's really based on a lot of faith and even though

scientists generally are opposed to intelligent design,

we all have some bias about the way natural laws were designed;

there's no question about it. You can talk to any practicing

physicist. We have a faith that underlying

laws of nature will have a certain elegance and a certain

beauty and a certain uniformity across all of natural phenomena.

That is a faith that we have. It's not a religious issue;

otherwise, I wouldn't bring it up in the classroom,

but it is certainly the credo of all scientists,

at least all physicists, that there is some elegance in

the laws of nature and we put a lot of money on that faith,

that the laws of nature will do this and will not do this.

Who are we to say that? Who are we to say nature

wouldn't have a system in which mechanics obeys the laws but

electricity and magnetism doesn't?

We haven't run into somebody called nature.

We don't worship a certain deity called nature but we

believe the laws of nature obey that.

So, even though scientists or physicists in particular may not

believe in design by any personal God,

they do believe in this underlying, rational system that

we are trying to uncover. You could be disproven,

you could be wrong in making the assumption,

but here it was right. It was really driven by this

notion that all laws of physics should obey the same principle

of relativity. So, Einstein's postulates are

that light behaves in this way because if it didn't behave in

this way, it would violate the principle

of relativity, whereas we know mechanical

phenomena do and electrical phenomena would not and that

cannot be the case. You have a question?

Student: Why would that not apply to the speed of sound?

Professor Ramamurti Shankar: Yes.

Because in the case of the speed of sound,

you can take the medium with you;

there is no such experiment you could perform.

See, in the train, if you could carry the ether

with you, there's no surprise you would get the same answer.

But we know we cannot carry the medium with us,

that comes from extra-terrestrial experiments.

That's why the velocity of sound is not elevated to a

fundamental velocity on which everybody will agree.

So, the two great postulates--You've got to know

where the postulates come from. Postulate Number 1 is simply a

restatement of the relativity principle.

I'll just say it in one sentence.

Exact wording is not important. All inertial observers are

equivalent.

"Equivalent" means each one of them is as privileged as any

other one to discover the laws of nature.

The laws of nature, we found, are not an accident

related to our state of motion. If I find some laws,

and you're moving relative to me, you'll find the same laws.

And if you and I find each other in relative motion,

you have as much right to claim you are at rest and I am moving

and I have as much right to claim that I am at rest and you

are moving. There is complete symmetry

between observers in uniform relative motion.

There is no symmetry between people in non-uniform [relative]

motion. As I said, non-uniform motion

creates effects which can destroy me and not destroy you.

So, no one's trying to talk their way out of acceleration,

whereas you can talk your way out of uniform velocity.

That's the first principle. So this was there even from the

time of Newton. What is true here is that all

inertial observers are equivalent with respect to all

natural phenomena, meaning all natural laws.

And that is a generalization, when we say "all" instead of

just mechanical.

And the Second Postulate, you call it a postulate because

there is just no way to deduce this,

is that the velocity of light is independent of the state of

motion of the source, of the observer, of everything.

If a light beam is emitted by a moving rocket,

it doesn't matter. If a light beam is seen in a

moving rocket, it doesn't matter.

All people will get the same answer for the velocity of

light. Student: Is there a

reason why the speed of light is constant?

Professor Ramamurti Shankar: No.

That's why it is a postulate. You can show a few things later

on. You can show that if there is

any other speed, which is the same for

everybody, that would have to be the speed of light.

In the final theory of relativity, there are not two or

three velocities that come out the same for everybody.

There is only one velocity that can have the same answer for all

people. That velocity is the velocity

of light. By that I don't mean it has to

be light itself. For example,

gravitational waves travel at the speed of light.

It's not just the light. It has to do with the velocity

of light being a single number which has to have the same value

for everybody. Okay, so it looks like he has

solved a big problem because he has said why light behaves this

way; light behaves this way because

it is part of the big conspiracy to hide uniform motion.

But you will see that you have made a terrible bargain because

once you take these two postulates,

you have restored the relativity principle to all

phenomena. Okay.

You've gone beyond mechanical phenomena to electro-magnetic

phenomena. But you will find that you have

to give up all the other cherished notions of Newtonian

physics. Think about why.

We are saying, here is a car going at 200

miles an hour, according to me.

You get into your own car and follow that car at 50 miles an

hour. You should get 150 but you keep

getting 200. This may not be true for cars

going at the speeds I mentioned but when finally you are talking

about a pulse of light, that is true.

And you've got to agree that is really not compatible with our

daily notions or with the formula I wrote down,

w = v - u. When you put v = c,

w has got to come out to be c.

And that's not a property of the Newtonian transformation.

So, what we are looking for is a new rule for connecting

x and t and x prime and t

prime, such that when the velocities

are computed and applied to the velocity of light,

you get the same answer. That's what we want to do now.

So, here is how we are going to do this.

Now, let's think about it. Let me send a pulse to the

right at speed c. You are going to the right at

three-fourths of c. My Newtonian expectation is,

you should get the speed of the pulse to be one-fourth of

c. But you insist it is c.

So, what will I say to you? What will I accuse you of doing?

I say you should get c/4 and you're getting c.

And you're finding velocity by finding the distance it travels

and dividing by the time so you are jacking up a number like

one-fourth c to c itself.

So, what could make you do that? Yep?

Student: If I perceive that you're moving forward,

or you're contracting your length,

then you're going to measure velocity keeping in mind you

should have had a greater length to begin with.

Professor Ramamurti Shankar: Yes.

So, one option--Let me repeat what he said.

I will say your meter sticks are being somehow shorter.

When you and I were buddies and were on the same train,

we agreed on the meter stick. But now we have gone on the

moving train. I will say there is something

wrong with your meter stick. Not only something wrong.

Specifically, I would say your meter sticks

have shrunk. For example,

if they have shrunk to one-fourth their size,

it is very clear that you would get a velocity of four times

what I expect. But there's another possibility.

Student: Time may be running slow.

Professor Ramamurti Shankar: Your clocks may be

running slow. So, you let the light travel

for four seconds and you thought it was only one second.

That's why you got four times the answer;

or it could be both. But something has to give.

And that is why it is an amazing theory.

That's why it is also amazing to me that somebody who was 26

years old would simply follow the consequence of this theory

and take it wherever it takes you but it is at the very

foundations of space and time that you have to modify.

So, even though you restored the relativity principle and

brought it back to the front, the price you have to pay is to

give up your notions that length is length and time is time.

We used to think a meter stick is a meter stick and a clock is

a clock. If I have a clock that ticks

out one second, you take it on a train,

I expect it to be ticking one second, but we're saying it's

not. So, something has to give in

length measurement or time measurement or both.

And that's what we're going to find out.

So, here's how you find that out.

Let us say that--Maybe I'll do it all on one blackboard because

this is the key to the whole calculation.

You remember now, if there is an event here,

you call it x prime and I call it x.

And according to me, you have traveled the distance

ut.

So, x prime equals x - ut is what we used to say

in the old days. And the converse of that is

x = x prime + ut. But now, we'll admit the fact

that maybe t and t prime are no longer the same.

But that's not all we will do. I will say, whenever you give

me a length x prime, I just don't buy it.

I take any length you give me and I jack it up by a factor of

γ [correction: should have said 1/γ]

to get the length according to me.

And you will take any answer I give you and you will multiply

it by the same factor. Don't worry about this yet!]

In other words, we don't buy our units of

length, so if you say it should be

x prime + ut prime, that's your formula

backwards for me. The coordinate of the event,

according to you, in the old days,

was x prime + ut. Now, we admit that t and

t prime may not be the same.

Then we say, but I will not take your

lengths, I will multiply them by γ to get the lengths

according to me. And you will not take my

expectations, but you will multiply it by the

same γ; that is the symmetry between

the two observers. In other words,

if I think your meter sticks are short and γ

[correction: should have said 1/γ]

is a number less than 1, I'm allowing you to accuse me

of having meter sticks which are short.

It is very interesting. If I said your meter sticks are

short and you say my meter sticks are longer than average,

that's an absolute difference. But we both accuse each other

of using shortened meter sticks, and so we use the same factor

of γ. We are going to find this

γ now. Student: How do you know

that both the distance and time are different?

Professor Ramamurti Shankar: We'll give it the

possibility that they are different and then we will see

that they are different. We know something's wrong with

space and something's wrong with time.

So, we'll not assume that t prime is equal to

t. You have to open up the

possibility. In the end, it may be that the

nature will say t prime is t and something

happens to length alone. But we'll find the answer is

more symmetric. Student: Why do we say

that the symmetry is the same? Professor Ramamurti

Shankar: Because that is the symmetry between the two

observers. If I want to say your meter

sticks are short, why should you concede that?

You should be able to accuse me of saying--The only difference

between you and me is you are moving to the right and I'm

moving to the left. Other than the sign of the

velocity, each person says the other person is moving and so we

will say that any length you give I'll discount by a factor

of γ. By symmetry,

anything I call a length, you will discount by the same

factor of γ. Now, let's apply this.

(x,t) was a certain event,

right? Let's apply it to the following

event. You have to follow this very

carefully. When you and I crossed,

remember that was at the origin of coordinates,

x equals to t equals to zero and x

prime equals to t prime equals to zero.

At that instant when our origins touched,

let us emit a flash of light. Okay, maybe when the origins

touched, there's a spark, the light signal goes out and

the light signal is detected here, by a light detector.

That second event, detection of the pulse,

it has a coordinates (x,t) for me;

it has a coordinates (x prime, t prime) for you.

The same event is given different coordinates.

We've already used the fact that coordinates will be

different but we are saying not only will x prime not be

equal to x; t prime may not be equal

to t either.

Okay, but now let's write down one important condition.

What is the relation between x prime and t

prime in this particular pair of events?

What is x prime? It is the location of the light

pulse after certain time? So, what's the relation of the

location of the detection of the light pulse and the time

t prime? Yes?

Student: [inaudible] Professor Ramamurti

Shankar: He is saying x prime--Do you guys

understand that? Do you agree with that

statement? This is not a random event.

The second event was the detection of a light pulse.

Light pulse left the origin t prime seconds earlier

and has come to this point, according to this guy,

and the ratio of the distance to the time is the velocity of

light. But it is also true that for

me, the light went to distance x in a time t.

So, that's the relation of x to t for me

also. I'm going to use those two

results and combine it with this to find this factor γ

and we will do that now. I want you to multiply the

left-hand side by the left-hand side and the right-hand side by

the right-hand side of that equation.

I hope you understand that in the Galilean days,

in the old days--Let's see what you will say.

I will say, x prime is x - ut because the

origins have shifted by an amount ut.

And you will say x is x prime plus ut

with the same time. Now, I'm saying time is

different. Not only that,

I don't buy your length. If you expect me to have this

length, I say "no." You exaggerated everything.

I'll scale it down by γ and vice versa.

So, if you multiply the left-hand side by the left-hand

side, you get xx prime, the right-hand side you get

γ^(2) times xx prime plus uxt prime

minus ux prime t minus u^(2)tt prime.

Now, divide everything by xx prime.

Then, you get 1 = γ^(2) times 1 + u.

If you divide this by xx prime, you'll get t prime

over x prime. If you divide this by xx

prime, you'll get t/x. And here you get u^(2)

times t/x, times t prime over

x prime.

So, what does that mean? Well, t prime over

x prime is 1/c and t/x is also

1/c because of what I wrote here.

If you want, let me write this as t

prime over x prime equals the c and t over

x equals the c. So, they will cancel.

And I'll get 1 = γ^(2) times 1 - u^(2) over

c^(2) because this t/x is a 1/c and

t prime over x prime is another 1/c.

So, that gives us the result that γ is 1 over square

root of 1 - u^(2) over c^(2).

If you plug the γ back in, you will find x prime

is x - ut divided by [square root of]

1 - u^(2) over c^(2).

Now, once you do that, once you got the relation

between x prime and x, you can go to the

lower equation and solve for t prime.

I don't feel like doing that. It's a simple algebraic

equation once you've got x prime how to solve for

t prime. Take the second of the two

equations and solve for t prime.

That detail I won't fill out, but you will get t prime

is t - ux over c^(2) divided by this.

So, I've not done every step but I've given you all the

things you need to do the one step.

There are equations up there that relate x and

x prime to t and t prime so if you can get

x prime in terms of x and t,

the other equation that you solve will give you t

prime in terms of x and t.

And this is the result.

Okay. This guy deserves two boxes

because it is the greatest result from relativity;

it's called the Lorentz transformation.

And we've been able to derive the Lorentz transformation with

what little we know. And you can see,

you can be a kid in high school and you can do this.

There's no calculus or anything else involved,

other than being open to the fact that the velocity of light

behaves in this strange way. Yes?

Student: How do you get t/x = c?

Professor Ramamurti Shankar: Why is t/x =

c? Oh, of course,

you're quite right. So, you caught the mistake here.

t/x is 1 over c and 1/c.

I really meant to write x = ct.

Yes? Student: [inaudible]

Professor Ramamurti Shankar: No.

If you define γ to be the absolute value by which you

transfer lengths from you to me, then you can take the positive

root. Student: [inaudible]

Professor Ramamurti Shankar: Well,

I can also tell you other reasons.

Let's take this formula here. Let's take the case where the

velocity is very small compared to the velocity of light.

That u/c is a very small number.

This number is almost 1 and I get x prime because x

- ut, which I know to be the correct answer at lower

velocities. If you pick the minus sign,

I'll get minus of x - ut, that's not the right

answer, not even close to the right answer.

At low velocities, if you go to velocity u

over c much less than 1, you have got to get back to

Galilean transformation. You can see if u over

c goes to zero. You can forget all about this

factor here. You get x prime is

x - ut and here, u over c can be

neglected, forget all that,

you get t prime equals t.

So, this coordinate transformation would reduce to

the Galilean transformation if the velocity between me and you

is much smaller than the velocity of light.

So, the formula really kicks in for velocities comparable to

velocity of light. Yes?

Student: [inaudible] What happens if u >

c? Professor Ramamurti

Shankar: Well, you start getting crazy

answers, right? You can already see that the

theory will not admit velocities bigger than the velocity of

light. You can already see it in this

formula. That tells you that the one

single velocity that you wanted to be the same for everybody is

also the greatest possible velocity;

that no observer can move at this speed with respect to

another observer that is equal to what is in excess of the

speed of light. So, the speed of light,

which came out to be a constant in the beginning of the theory,

is also turning out to be the upper limit on possible

velocities. That's the origin of the

statement that no observer can travel at a speed bigger than

light, but we'll discuss it more and more.

But you have to understand what it is that is being derived,

what is the meaning of this formula here.

What is it telling you? If I say, this is called the

Lorenz transformations, what do they tell you?

What are these numbers and what's their significance?

Would you like to try? Student: Well,

one thing that they tell you is if u happens to be

greater than c [inaudible]

Professor Ramamurti Shankar: No,

no, I don't mean what happens in the formula in special cases.

What is it relating? What is xt and what is

x prime t prime? Student:

Okay, so x prime would be the distance that the person

who's traveling at the higher speed experiences.

And [inaudible] and so for that person,

distance is going to seem shorter?

Professor Ramamurti Shankar: No.

See, I'm not even telling you to get consequences of the

equation. What are the numbers x

and t in this equation? And what are the numbers

x prime and t prime in this equation?

Student: x and t are distance and time

for a person who is in an inertial frame of reference,

who is not moving. Professor Ramamurti

Shankar: Right. Student: And x

prime and t prime are distance and time for the person

who is moving at the speed of [inaudible]

Professor Ramamurti Shankar: And when you say

distance and time, what do you mean,

distance and time? Student: The way that

the length of the distance will seem to them that they travel.

Professor Ramamurti Shankar: But what is

happening at xt; it's the coordinates of what?

Student: Of their location [inaudible]

Professor Ramamurti Shankar: They are located at

zero, zero. Right?

What's happening at x and t?

Yes? Student: It's observing

the event. Professor Ramamurti

Shankar: It's the event. The key I was looking for is an

event. You've got to understand what

the formula is connecting. Things are happening in space

and in time, right? Something happens here.

That something has a spatial coordinate and a time

coordinate, according to two observers.

The observers originally had their origins and their clocks

coincide when they passed; that's how they're related.

And one is moving to the right at speed u.

Then, the claim is that if one event had a coordinate x

and t for one person, for the other person moving to

the right at speed u, the same event would have

coordinates x prime and t prime and the relation

between x and t and x prime and t

prime is this. Yes?

Student: The two observers [inaudible]

don't they observe different laws and [inaudible]

Professor Ramamurti Shankar: No.

The fact that an event has different coordinates doesn't

mean that you are observing different laws.

For example, let's take that fire

extinguisher. We look at it,

it's obeying F = ma, right?

The coordinates of the fire extinguisher with me as the

origin is quite different from you as the origin.

Student: [inaudible] Professor Ramamurti

Shankar: You mean in these new equations?

Yes, in these new equations, F = ma will not work;

that's correct. Student: They are

inertial references, even though they are moving?

Professor Ramamurti Shankar: Ah.

Yes. So, the point is the laws of

Newton themselves have to be modified.

F = ma will be modified in a certain way but the new

modified laws will reduce to F = ma at low velocities,

which is why in the old days it looked like F = ma.

But there will be new laws, but they will also have the

property that when I measure them I'll get the laws that will

agree with what you measure. Yes?

Student: Does individual values for time or distance

still have to agree? Professor Ramamurti

Shankar: The coordinates of an event will differ from person

to person. That's not the same as saying

the laws as deduced will be different.

For example, there are two stars which are

attracted to each other by gravitation and they are

orbiting around their common center of mass.

If I see them, I will find that they obey the

law of gravitation with m_1m

_2 over r^(2) where r is

the distance between the points and the acceleration is whatever

I think it is. You can go on a rocket and look

at the same two stars. They will be in a more

complicated motion, maybe the whole system will be

drifting a little to you, but their acceleration will be

the same as what I get and the force between them will also be

the same as what I get and the laws that you would deduce by

looking at that star would be the same laws that I would

deduce. So, that's a difference between

the laws being the same and the coordinates being the same.

No one said x prime and x are the same in that

equation there. They are different.

We are looking at it from different vantage points.

But the fact is that force is equal to mass times acceleration

is the same for the two people. Okay.

The laws will be the same but things won't look the same.

For example, you can stand on your head,

don't even have to go to another frame of reference;

you can stand on your head, your z coordinate is the

minus of my z coordinate; to every point I give a

z, you will give a minus z.

But the world, even though you are a little

messed up and want to stand on your head,

you have every right to do that and you will find that F =

ma. So the point is,

the way we see events may depend on their origin or

coordinates, but the laws we deduce are to

be distinguished from the perception that we have.

Okay? For example,

if I'm on the ground, I send a piece of chalk,

it goes up and it goes down. If you see me from a moving

train, you would think it went on a parabola.

So, no one says the chalk will go up and down for you.

For you, it will go like this but its motion will still obey

F = ma, is what I'm saying.

That's all you really mean by saying things look the same.

So, what you have to understand is that Lorentz transformations

are the way to relate a pair of events, given events.

Here's a simple example. If you live in the xy

plane, there's a point here. It's not an event,

it's simply a point in the xy plane.

You measure it this way and that way and you call it the

coordinates. If somebody else picks a

different coordinate system with an angle θ,

that person will say that's x prime and that is

y prime and x prime and y prime are not

the same. I remind you,

x prime is x cos> θ [delete "minus

y"] plus y sin θ,

etc., and y prime is something else.

θ is the angle between the two observers.

So, the point is the point. It certainly looks different to

the two people, but the same point has two

coordinates. Similarly, the same event,

like the collision of two cars, will have different events for

different people. That's not the new part.

The new part is that the rules for connecting xt to

x't', is quite different from the Galilean rules,

new rules. It's what you guys have to

understand. And finally,

why did Einstein get the credit for turning the world into four

dimensions instead of three? After all, x and

t were present there, too.

The point is, t prime is always equal

to t no matter how you move,

whereas in the Einstein theory, x and t get

scrambled into x prime and t prime just the way

x and y get scrambled into each other when

you rotate your axis. So, to have time as another

variable that doesn't transform at all is not the same as making

it into a coordinate. The four-dimensional world of

Einstein is four-dimensional because space and time mix with

each other when you change your frame of reference.

That's what makes t now a coordinate as previously it

was something the same for all people.